Discovering Frequent Episodes With General Partial Orders

Abstract

Pattern Discovery, a popular paradigm in data mining refers to a class of techniques that try and extract some unknown or interesting patterns from data. The work carried out in this thesis concerns frequent episode mining, a popular framework within pattern discovery, with applications in alarm management, fault analysis, network reconstruction etc. The data here is in the form of a single longtime-ordered stream of events. The pattern of interest here, namely episode, is basically a set of event-types with a partial order on it. The task here is to unearth all patterns( episodes here) which have a frequency above a user-defined threshold irrespective of pattern size. Most current discovery algorithms employ a level-wise a priori-based method for mining, which basically adopts a breadth-first search strategy of the space of all episodes. The episode literature has seen multiple ways of defining frequency with each definition having its own set of merits and demerits. The main reason for different frequencies definitions being proposed is that, in general, counting all occurrences of a set of episodes is computationally very expensive. The first part of the thesis gives a unified view of all the apriori-based discovery algorithms for serial episodes(associated with a total order)under these various frequencies. Specifically, the various existing counting algorithms can be viewed as minor modifications of each other. We also provide some novel proofs of correctness for some of the serial episode counting schemes, which in turn can be generalized to episodes with general partial orders. Our unified view helps us derive quantitative relationships between different frequencies. We also discuss all the anti-monotonicity properties satisfied by the various frequencies, a crucial information needed for the candidate generation step. The second part of the thesis proposes discovery algorithms for episodes with general partial orders, for which no algorithms currently exist in literature. The discovery algorithm proposed is apriori-based and generalizes the existing serial and parallel (associated with a trivial order) episode algorithms. The discovery algorithm is a level-wise procedure involving the steps of candidate generation and counting a teach level. In the context of general partial orders, a major problem in a priori-based discovery is to have an efficient candidate generation scheme. We present a novel candidate generation algorithm for mining episodes with general partial orders. The counting algorithm design for general partial order episodes draws ideas from the unified view of counting for serial episodes, presented in the first part of the work. We formally show the correctness of the proposed candidate generation and counting steps for general partial orders. The proposed candidate generation algorithm is flexible enough to be able to mine in certain specialized classes of partial orders (satisfying what we call maximal sub episode property), of which, the serial and parallel class of episodes are two specific instances. Our algorithm design initially restricts itself to the class of general partial order episodes called injective episodes wherein repeated event-types are not allowed. We then generalize this to a larger class of episodes called chain episodes, where episodes can have some repeated event types. The class of chain episodes contains all (including non-injective) serial and parallel episodes and thus our method properly generalizes the existing methods for serial and parallel episode discovery. We also discuss some problems in extending our algorithms to episodes beyond the class of chain episodes. Also, we demonstrate that frequency alone is not a sufficient enough interestingness measure for episodes with unrestricted partial orders. To address this issue, we propose an additional measure called bidirectional evidence to assess interestingness which, along with frequency is found to be extremely effective in unearthing interesting patterns. In the frequent episode framework, the choice of thresholds are most often user-defined and arbitrary. To address this issue, the last part of the work deals with assessing significance of partial order episodes in a statistical sense based on ideas from classical hypothesis testing. We declare an episode to be significant if its observed frequency in the data stream is large enough to be very unlikely, under a random i.i.d model .The key step in the significance analysis involves the mean and variance computation of the the time between successive occurrences of the pattern. This computation can be reformulated as, solving for the mean and variance of the first visit time to a particular stat e in an associated Markov chain. We use a generating function approach to solve for this mean and variance. Using this and a Gaussian approximation to the frequency random variable, we can now calculate a frequency threshold for any partial order episode, beyond which we infer it to be significant. Our significance analysis for general partial order episodes generalizes the existing significance analysis of serial episode patterns. We demonstrate on synthetic data the effectiveness of our significance thresholds.